A fast iterative scheme for the linearized Boltzmann equation

Lei Wu, Jun Zhang, Haihu Liu, Yonghao Zhang, Jason M. Reese

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Iterative schemes to find steady-state solutions to the Boltzmann equation is efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator $L$ into the form $L=(L+N\delta{h})-N\delta{h}$, where $\delta$ is the gas rarefaction parameter, $h$ is the velocity distribution function, and $N$ is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when $N\delta$ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.
Language English 431–451 21 Journal of Computational Physics 338 7 Mar 2017 10.1016/j.jcp.2017.03.002 Published - 1 Jun 2017

Fingerprint

Boltzmann equation
Transpiration
continuum flow
transpiration
Velocity distribution
rarefaction
Distribution functions
Flow of gases
preserving
collisions
gas flow
Lennard-Jones potential
velocity distribution
distribution functions
operators
Gases
Gas mixtures
Flow velocity
rarefied gases
Porous materials

Keywords

• linearized Boltzmann equation
• rarefied gas dynamics
• synthetic iterative scheme
• Lennard-Jones potential
• gas mixture

Cite this

Wu, Lei ; Zhang, Jun ; Liu, Haihu ; Zhang, Yonghao ; Reese, Jason M. / A fast iterative scheme for the linearized Boltzmann equation. In: Journal of Computational Physics. 2017 ; Vol. 338. pp. 431–451.
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title = "A fast iterative scheme for the linearized Boltzmann equation",
abstract = "Iterative schemes to find steady-state solutions to the Boltzmann equation is efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator $L$ into the form $L=(L+N\delta{h})-N\delta{h}$, where $\delta$ is the gas rarefaction parameter, $h$ is the velocity distribution function, and $N$ is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when $N\delta$ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.",
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author = "Lei Wu and Jun Zhang and Haihu Liu and Yonghao Zhang and Reese, {Jason M.}",
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A fast iterative scheme for the linearized Boltzmann equation. / Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.

In: Journal of Computational Physics, Vol. 338, 01.06.2017, p. 431–451.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A fast iterative scheme for the linearized Boltzmann equation

AU - Wu, Lei

AU - Zhang, Jun

AU - Liu, Haihu

AU - Zhang, Yonghao

AU - Reese, Jason M.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - Iterative schemes to find steady-state solutions to the Boltzmann equation is efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator $L$ into the form $L=(L+N\delta{h})-N\delta{h}$, where $\delta$ is the gas rarefaction parameter, $h$ is the velocity distribution function, and $N$ is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when $N\delta$ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.

AB - Iterative schemes to find steady-state solutions to the Boltzmann equation is efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator $L$ into the form $L=(L+N\delta{h})-N\delta{h}$, where $\delta$ is the gas rarefaction parameter, $h$ is the velocity distribution function, and $N$ is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when $N\delta$ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.

KW - linearized Boltzmann equation

KW - rarefied gas dynamics

KW - synthetic iterative scheme

KW - Lennard-Jones potential

KW - gas mixture

UR - http://www.sciencedirect.com/science/journal/00219991

U2 - 10.1016/j.jcp.2017.03.002

DO - 10.1016/j.jcp.2017.03.002

M3 - Article

VL - 338

SP - 431

EP - 451

JO - Journal of Computational Physics

T2 - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -